| L(s) = 1 | − 1.73·3-s − 5.27·11-s − 2.62·13-s − 0.418·17-s − 3.27·19-s − 7.82·23-s + 5.19·27-s + 4.27·29-s − 3.27·31-s + 9.13·33-s − 9.97·37-s + 4.54·39-s + 3.72·41-s − 2.15·43-s − 6.50·47-s + 0.725·51-s − 5.67·53-s + 5.67·57-s + 3.27·59-s + 13.5·61-s + 3.52·67-s + 13.5·69-s − 4.54·71-s − 6.50·73-s + 7.27·79-s − 9·81-s + 7.40·83-s + ⋯ |
| L(s) = 1 | − 1.00·3-s − 1.59·11-s − 0.728·13-s − 0.101·17-s − 0.751·19-s − 1.63·23-s + 1.00·27-s + 0.793·29-s − 0.588·31-s + 1.59·33-s − 1.63·37-s + 0.728·39-s + 0.581·41-s − 0.327·43-s − 0.949·47-s + 0.101·51-s − 0.779·53-s + 0.751·57-s + 0.426·59-s + 1.73·61-s + 0.430·67-s + 1.63·69-s − 0.539·71-s − 0.761·73-s + 0.818·79-s − 81-s + 0.812·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3636776387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3636776387\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 13 | \( 1 + 2.62T + 13T^{2} \) |
| 17 | \( 1 + 0.418T + 17T^{2} \) |
| 19 | \( 1 + 3.27T + 19T^{2} \) |
| 23 | \( 1 + 7.82T + 23T^{2} \) |
| 29 | \( 1 - 4.27T + 29T^{2} \) |
| 31 | \( 1 + 3.27T + 31T^{2} \) |
| 37 | \( 1 + 9.97T + 37T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 + 2.15T + 43T^{2} \) |
| 47 | \( 1 + 6.50T + 47T^{2} \) |
| 53 | \( 1 + 5.67T + 53T^{2} \) |
| 59 | \( 1 - 3.27T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 3.52T + 67T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 + 6.50T + 73T^{2} \) |
| 79 | \( 1 - 7.27T + 79T^{2} \) |
| 83 | \( 1 - 7.40T + 83T^{2} \) |
| 89 | \( 1 - 7T + 89T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.171257493827192608437804504490, −7.56982007761850293306538599981, −6.66041763826977995963316918660, −6.06748394932736942386443176166, −5.20570894903667854883256791398, −4.93572503787074097524276285394, −3.84488628235524053666349281441, −2.74077712759499915437497592392, −1.96667246598051312791277239387, −0.32561725763582843796863911732,
0.32561725763582843796863911732, 1.96667246598051312791277239387, 2.74077712759499915437497592392, 3.84488628235524053666349281441, 4.93572503787074097524276285394, 5.20570894903667854883256791398, 6.06748394932736942386443176166, 6.66041763826977995963316918660, 7.56982007761850293306538599981, 8.171257493827192608437804504490
